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626 results on '"MARKOWICH, PETER A."'

164. On uniform decay of the entropy for reaction-diffusion systems

Author

Mielke, Alexander, Haskovec, Jan, and Markowich, Peter A.

Subjects
mass-action law, 35K57, 35B40, log-Sobolev inequality, 92E20, exponential decay of relative entropy, Reaction-diffusion
Abstract

In this work we derive entropy decay estimates for a class of nonlinear reaction-diffusion systems modeling reversible chemical reactions under the assumption of detailed balance. In particular, we provide explicit bounds for the exponential decay of the relative logarithmic entropy, being based essentially on the application of the log-Sobolev inequality and a convexification argument only, making it quite robust to model variations. An important feature of our analysis is the interaction of the two different dissipative mechanisms: pure diffusion, forcing the system asymptotically to the homogeneous state, and pure reaction, forcing the solution to the (possibly inhomogeneous) chemical equilibrium. Only the interaction of both mechanisms provides the convergence to the homogeneous equilibrium. Moreover, we introduce two generalizations of the main result: we allow for vanishing diffusion constants in some chemical components, and we consider different entropy functionals. We provide a few examples to highlight the usability of our approach and shortly discuss possible further applications and open questions.

Published
2013
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165. Fundamental solutions for Schrödinger operators with general inverse square potentials.

Author

Chen, Huyuan, Alhomedan, Suad, Hajaiej, Hichem, and Markowich, Peter

Subjects
SCHRODINGER equation, LINEAR operators, GRAPH theory, INVERSE functions, MATHEMATICAL analysis
Abstract

In this paper, we clarify the fundamental solutions for Schrödinger operators given as , where the potential V is a general inverse square potential in with . In particular, letting ,where , we discuss the existence and nonexistence of positive fundamental solutions for Hardy operator , which depend on the parameter t. [ABSTRACT FROM AUTHOR]

Published
2018
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166. An analysis of quantum Fokker-Planck models : a Wigner function approach

Author

Arnold, Anton, López, José L., Markowich, Peter A., and Soler, Juan

Subjects
Wigner equation, 35S10, open quantum system, self-similarity, 35Q40, 81Q99, large-time behavior, 81V99
Abstract

The analysis of dissipative transport equations within the framework of open quantum systems with Fokker-Planck-type scattering is carried out from the perspective of a Wigner function approach. In particular, the well-posedness of the self-consistent whole-space problem in 3D is analyzed: existence of solutions, uniqueness and asymptotic behavior in time, where we adopt the viewpoint of mild solutions in this paper. Also, the admissibility of a density matrix formulation in Lindblad form with Fokker-Planck dissipation mechanisms is discussed. We remark that our solution concept allows to carry out the analysis directly on the level of the kinetic equation instead of on the level of the density operator.

Published
2011
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172. Nonlinear diffusions as limit of kinetic equations with relaxation collision kernels

Author

Dolbeault, Jean, Markowich, Peter, Oelz, Dietmar, Schmeiser, Christian, CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Faculty of Mathematics [Vienna], University of Vienna [Vienna], Institute for Analysis and Scientific Computing [Wien], Vienna University of Technology (TU Wien), Johann Radon Institute for Computational and Applied Mathematics (RICAM), and Austrian Academy of Sciences (OeAW)

Subjects
Boltzmann equation, Fast diffusion equation, 82C40, 35B40, 35A05, 35K55, 45K05, 82D05, 85A05, Relaxation time approximation, Gibbs state, Equilibrium distribution function, Kinetic equation, Compensated compactness, Diffusion limit, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Macroscopic limit, Porous medium equation
Abstract

International audience; Kinetic transport equations with a given confining potential and non-linear relaxation type collision operators are considered. General (monotone) energy dependent equilibrium distributions are allowed with a chemical potential ensuring mass conservation. Existence and uniqueness of solutions is proven for initial data bounded by equilibrium distributions. The diffusive macroscopic limit is carried out using compensated compactness theory. The result are drift-diffusion equations with nonlinear diffusion. The most notable examples are of porous medium or fast diffusion type, with exponent ranging from 0 to 5/3, in dimension 3.

Published
2007

180. Orbital stability of standing waves of a class of fractional Schrödinger equations with Hartree-type nonlinearity.

Author

Cho, Yonggeun, Fall, Mouhamed M., Hajaiej, Hichem, Markowich, Peter A., and Trabelsi, Saber

Subjects
STANDING waves, STABILITY (Mechanics), SCHRODINGER equation, NONLINEAR theories, EXISTENCE theorems
Abstract

This paper is devoted to the mathematical analysis of a class of nonlinear fractional Schrödinger equations with a general Hartree-type integrand. We show the well-posedness of the associated Cauchy problem and prove the existence and stability of standing waves under suitable assumptions on the nonlinearity. Our proofs rely on a contraction argument in mixed functional spaces and the concentration-compactness method. [ABSTRACT FROM AUTHOR]

Published
2017
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189. SIAM Journal on Control and Optimization / Optimal bilinear control of Gross-Pitaevskii equations

Author

Hintermüller, Michael, Hintermüller, Michael, Marahrens, Daniel, Markowich, Peter A., Sparber, Christof, Hintermüller, Michael, Hintermüller, Michael, Marahrens, Daniel, Markowich, Peter A., and Sparber, Christof

Abstract

A mathematical framework for optimal bilinear control of nonlinear Schrödingerequations of Gross–Pitaevskii type arising in the description of Bose–Einstein condensates is presented.The obtained results generalize earlier efforts found in the literature in several aspects. Inparticular, the cost induced by the physical workload over the control process is taken into accountrather than the often used L2- or H1-norms for the cost of the control action. Well-posedness ofthe problem and existence of an optimal control are proved. In addition, the first order optimalitysystem is rigorously derived. Also a numerical solution method is proposed, which is based on aNewton-type iteration, and used to solve several coherent quantum control problems.

Published
2013
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198. Kinetic relaxation models for energy transport

Author

60271011, Aoki, Kazuo, Markowich, Peter, Takata, Shigeru, 60271011, Aoki, Kazuo, Markowich, Peter, and Takata, Shigeru

Abstract

Kinetic equations with relaxation collision kernels are considered under the basic assumption of two collision invariants, namely mass and energy. The collision kernels are of BGK-type with a general local Gibbs state, which may be quite different from the Gaussian. By the use of the diffusive length/time scales, energy transport systems consisting of two parabolic equations with the position density and the energy density as unknowns are derived on a formal level. The H theorem for the kinetic model is presented, and the entropy for the energy transport systems, which is inherited from the kinetic model, is derived. The energy transport systems for specific examples of the global Gibbs state, such as a power law with negative exponent, a cut-off power law with positive exponent, the Maxwellian, Bose–Einstein, and Fermi–Dirac distributions, arepresented.

Published
2007

200. On Uniform Decay of the Entropy for Reaction-Diffusion Systems.

Author

Mielke, Alexander, Haskovec, Jan, and Markowich, Peter

Subjects
ENTROPY (Information theory), EXPONENTIAL decay law, REACTION-diffusion equations, SOBOLEV spaces, CHEMICAL equilibrium
Abstract

This work provides entropy decay estimates for classes of nonlinear reaction-diffusion systems modeling reversible chemical reactions under the detailed-balance condition. We obtain explicit bounds for the exponential decay of the relative logarithmic entropy, being based essentially on the application of the Log-Sobolev estimate and a convexification argument only, making it quite robust to model variations. An important feature of our analysis is the interaction of the two different dissipative mechanisms: pure diffusion, forcing the system asymptotically to the homogeneous state, and pure reaction, forcing the solution to the (possibly inhomogeneous) chemical equilibrium. Only the interaction of both mechanisms provides the convergence to the homogeneous equilibrium. Moreover, we introduce two generalizations of the main result: (i) vanishing diffusion constants in some chemical components and (ii) usage of different entropy functionals. We provide a few examples to highlight the usability of our approach and shortly discuss possible further applications and open questions. [ABSTRACT FROM AUTHOR]

Published
2015
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